A SPectroscopic Survey of Biased Halos in the Reionization Era (ASPIRE): Spectroscopically Complete Census of Obscured Cosmic Star Formation Rate Density at z = 4-6

We present a stringent measurement of the dust-obscured star formation rate density (SFRD) at z = 4-6 from the ASPIRE JWST Cycle-1 medium and ALMA Cycle-9 large program. We obtained JWST/NIRCam grism spectroscopy and ALMA 1.2 mm continuum map along 25 independent quasar sightlines, covering a total survey area of ∼35 arcmin² where we search for dusty star-forming galaxies (DSFGs) at z = 0-7. We identify eight DSFGs in seven fields at z = 4-6 through the detection of Hα or [O iii] λ5008 lines, including fainter lines such as Hβ, [O iii] λ4960, [N ii] λ6585, and [S ii] λλ6718,6733 for six sources. With this spectroscopically complete DSFG sample at z = 4-6 and negligible impact from cosmic variance (shot noise), we measure the infrared luminosity function (IRLF) down to L_IR ∼ 2 × 10¹¹ L_⊙. We find flattening of IRLF at z = 4-6 towards the faint end (power-law slope α = 0.5 9 − 0.45 + 0.39 ). We determine the dust-obscured cosmic SFRD at this epoch to be log [ ρ SFR , IR / ( M ⊙ yr − 1 Mpc − 3 ) ] = − 1.5 2 − 0.13 + 0.14 . This is significantly higher than previous determinations using ALMA data in the Hubble Ultra Deep Field, which is void of DSFGs at z = 4-6 because of strong cosmic variance (shot noise). We conclude that the majority (66% ± 7%) of cosmic star formation at z ∼ 5 is still obscured by dust. We also discuss the uncertainty of SFRD propagated from far-IR spectral energy distribution and IRLF at the bright end, which will need to be resolved with future ALMA and JWST observations.